In this note we consider subfamilies of the ideal s0 introduced by Marczewski-Szpilrajn and ideals sp0, l0 analogously defined using complete Laver trees and Laver trees respectively. We show that under some set-theoretical assumptions =c\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$${cov=\mathfrak{c}}$$\end{document} for example) in every uncountable Polish space X every family A⊆s0\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$${\mathcal{A}\subseteq s_0}$$\end{document} covering X has a subfamily with s-nonmeasurable union. We show the consistency of cov=ω1 (...) \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$${cov=\omega_1 < \mathfrak{c}}$$\end{document} with the existence of a partition A∈[s0]ω1\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$${\mathcal{A}\in[s_0]^{\omega_1}}$$\end{document} of the real line with a subfamily A′⊆A\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$${\mathcal{A}'\subseteq\mathcal{A}}$$\end{document} for which ⋃A′\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$${\bigcup\mathcal{A}'}$$\end{document} is s-nonmeasurable. We also show that it is relatively consistent with ZFC that ω1shrink)

In this paper we study a notion of a κ -covering set in connection with Bernstein sets and other types of non-measurability. Our results correspond to those obtained by Muthuvel in [7] and Nowik in [8]. We consider also other types of coverings.

A \(\sigma \) -ideal \(\mathcal {I}\) on a Polish group \((X,+)\) has the Smital Property if for every dense set _D_ and a Borel \(\mathcal {I}\) -positive set _B_ the algebraic sum \(D+B\) is a complement of a set from \(\mathcal {I}\). We consider several variants of this property and study their connections with the countable chain condition, maximality and how well they are preserved via Fubini products. In particular we show that there are \(\mathfrak {c}\) many maximal invariant \(\sigma (...) \) -ideals with Borel bases on the Cantor space \(2^\omega \). (shrink)

The two‐dimensional version of the classical Mycielski theorem says that for every comeager or conull set there exists a perfect set such that. We consider a strengthening of this theorem by replacing a perfect square with a rectangle, where A and B are bodies of some types of trees with. In particular, we show that for every comeager Gδ set there exist a Miller tree and a uniformly perfect tree such that and that cannot be a Miller tree. In the (...) case of measure we show that for every subset F of of full measure there exists a uniformly perfect tree such that and no side of such a rectangle can be a body of a Silver tree or a Miller tree. We also show some properties of forcing extensions from which we derive nonstandard proofs of Mycielski‐like theorems via the Shoenfield Absoluteness Theorem. (shrink)

In this paper, we consider a notion of nonmeasurablity with respect to Marczewski and Marczewski-like tree ideals $s_0$, $m_0$, $l_0$, $cl_0$, $h_0,$ and $ch_0$. We show that there exists a subset of the Baire space $\omega ^\omega,$ which is s-, l-, and m-nonmeasurable that forms a dominating m.e.d. family. We investigate a notion of ${\mathbb {T}}$ -Bernstein sets—sets which intersect but do not contain any body of any tree from a given family of trees ${\mathbb {T}}$. We also obtain a (...) result on ${\mathcal {I}}$ -Luzin sets, namely, we prove that if ${\mathfrak {c}}$ is a regular cardinal, then the algebraic sum of a generalized Luzin set and a generalized Sierpiński set belongs to $s_0, m_0$, $l_0,$ and $cl_0$. (shrink)

We show that under some conditions on a family A ⊂ I there exists a subfamily A0 ⊂ A such that ∪ A0 is nonmeasurable with respect to a fixed ideal I with Borel base of a fixed uncountable Polish space. Our result applies to the classical ideal of null subsets of the real line and to the ideal of first category subsets of the real line.